Under the restrictions that the mapping functions of transformation are defined in extended two-dimensional (2D) forms and the incident waves are 2D propagating fields, the conditions for non-reflecting boundaries in a finite-embedded coordinate transformation metamaterial slab are derived. By exploring several examples, including some reported in the literatures and some novel ones developed in this study, we show that our approach can be efficiently used to determine the condition for a finite-embedded coordinate transformed metamaterial slab to be non-reflecting.
©2010 Optical Society of America
Coordinate transformation optics (hereafter referred to as transformation optics) is a recently introduced concept of tailoring electromagnetic waves to achieve certain desirable characteristics via the use of specially designed metamaterials . The technique is based on the form-invariance of Maxwell’s equations under smooth and differentiable coordinate transformations . The mathematically defined transformations are cast into corresponding changes of constitutive material parameters and consequently, of electromagnetic entities, such as electromagnetic fields, currents, and charge distributions. The invisibility cloak has been proposed  as an innovative example of these, and many related experimental  and theoretical studies have been proposed. In addition to cloaking devices, including the carpet cloak , acoustic cloak , matter wave cloak  and an anti-cloak , other interesting concepts of coordinate transformation optics such as adaptive beam bends and beam expanders , field concentrators , source transformation devices  and illusion optical devices  have been reported as well.
From a topological view point, transformation optics can be categorized into two types. In the first type of transformation, the spatial domain of the transformation is ideally localized within a finite space: in other words, the interaction of light waves with the transformed region is designed to occur near localized objects. The invisibility cloak (free space), carpet cloak (ground mirror) or illusion optics (other objects) are such examples. Therefore, the properties of outgoing light waves from such transformed media, at least in ideal cases, recover those of their original states. In contrast, the other type of transformation optical device has been proposed, i.e. finite-embedded coordinate transformation optical devices . In this type of transformation, the metrics of space are not necessarily recovered into those of the original space. In many cases, the main objective of transformed optical media is to modify the outgoing wavefronts. The beam shifter, divider  and expander  are examples of this type.
The concept of finite-embedded coordinate transformation optics has stimulated some interest in the reflection conditions at the interfaces between transformed and untransformed media. Due to some slight vagueness in the non-reflecting condition between these media suggested in the original introductory paper , the issue of the necessary and/or sufficient conditions for finite-embedded media to be transparent has been discussed [8,13–17]. The non-reflecting condition derived in Ref . covers three-dimensionally (3D) transformed media with non-planar boundaries. The derived condition given in Eq. (11) in Ref. . states that if the boundary coordinate in the transformed medium can be expressed as the rotation of the original space with a constant displacement vector added, the boundary is non-reflecting. However, in this report we will show that a non-reflecting coordinate transformed boundary can exist for a skewed-and-expanded transformation (see Section 3.2 and Fig. 4 ) which cannot be covered by Eq. (11) in Ref. . Another general study on the boundary conditions for generalized transformed media was also reported . Although our study is restricted to more specific cases of extended two-dimensional (2D) problems, we provide more explicit conditions and illustrative examples throughout this paper which were not addressed in Ref. . Recently, studies on the mapping method that can make the 2D transformation optical slab transparent in a general way have been reported [15–17]. However, the discussions concerning this are limited only to cases of 2D transformed media and cannot deal with extended 2D cases which are treated in our paper (see Section 4 and Fig. 5 ).
In this work, the non-reflecting boundary conditions of a finite-embedded coordinate transformed metamaterial slab will be explored. For the sake of simplicity, we imposed two restrictions in our study. The first one is that the mapping functions of the coordinate transformation, and are not z-dependent: only can be x-, y- and z-dependent. This is more general than the 2D transformations treated in Refs. [15–17]. As will be seen in this study, this is not a straightforward extension of the 2D case and care must be taken in deriving the non-reflecting condition. We also assume that the incident field is defined in 2D conventions, i.e., we exclude conical incidence. With this assumption, the dispersion relation in the transformed media can be given in a very simple form. The governing equations which make transformed media transparent are presented, along with details of the derivation procedure. Several examples, including known mapping functions in several literatures as well as some novel ones, will be discussed with numerical implementations. A discussion of the design of some novel transformation optical devices is also presented.
2. Derivation of the governing equation
In this section, we start with the basics of the coordinate transformation in a Cartesian coordinate system and derive an equation which describes the non-reflecting boundaries in 2D finite-embedded coordinate transformation media (CTM). Let us consider a coordinate transformation in a Cartesian coordinate system, which maps the points in the original space (or the virtual space) to those in the physical space (or the transformed space) as follows:Eq. (1) can have inverse transformations.
Now let us consider the case where this CTM is finitely embedded as in Fig. 1 . We assume that the material parameters in the original space can be different from those of the surrounding media in the physical space. As depicted in Fig. 1, the transformed medium (region II) fills a finite-extent (along x’-direction) of the physical space and the material parameters of both sides of the transformed medium (region I and III) can also be different.
For the sake of simplicity, we further assume that the mapping functions in Eq. (1) are defined in a sort of extended 2D convention such that
This means that the mapping functions and are not z-dependent but can be x- and y-dependent by the scale factor : this is the reason why we refer to this mapping as extended 2D case.
From the form-invariance of Maxwell’s equations, the material tensors in virtual space can be re-expressed in the physical space as with the following relation :
Throughout this paper, we assume that the material in the virtual space is isotropic and homogeneous so that the material parameters in the original space can be expressed asand. Under this condition, the material tensor in transformed space can be defined as and with where A is the Jacobian matrix and the superscript t denotes the matrix transpose.
Now suppose that the material tensor can be expressed asEq. (1). Substituting Eq. (5) into Maxwell’s equations with time dependency of gives the following wave equation in a matrix form:Eq. (8) is given asEq. (8) has a non-trivial solution, the determinant of, where I is the identity matrix, should be zero. Therefore, we arrive at the following dispersion relation for this transformed media
For simplicity, we assume that the incident plane wave arising from region I has . Under this assumption and for a non-zero n, the dispersion relation in Eq. (10) can be expressed with cofactor elements as:
In general, the transformed medium in region II is inhomogeneous and anisotropic and thus, cannot be approximated efficiently using simple models such as a stair-case approximation. Therefore, we restrict our analysis to those cases in which the variations in the material parameters in the transformed space are smooth and differentiable. Furthermore, since our interest is to attempt to determine conditions that make this finite slab of CTM non-reflecting, multiple reflections can be ignored. As a consequence, we can consider the reflections at each interface ( and ) separately as if they were single interfaces. In addition, when we obtain the non-reflecting condition at the first boundary (), the condition required for reflectance of the second boundary () to be zero can be obtained very easily, i.e., we can adopt the condition obtained at the first boundary with an appropriate change in material parameters from I to III.
From an analysis of eigenvalues and eigenvectors of this transformed media under the assumption of a plane monochromatic wave, it is apparent that the TE- and TM-polarized fields are eigen-bases of this transformed material. Therefore, we can decompose the problem into TE- and TM-polarized cases. Now let us consider the reflectance of the TM-polarized plane monochromatic wave [and] incident on the first boundary. The magnetic field components in the immediate vicinity of the first boundary are readily expressed as
The tangential electric field in region II can be obtained similarly with the material parameters given in Eq. (5) as
From the continuity of the tangential components of the electromagnetic field at the boundary, we have
Therefore, to make r zero, we have
To make the medium non-reflecting for all incidence angles (i.e. for arbitrary values of ), Eq. (18) should be satisfied for the arbitrary values of. Considering Eqs. (11) and (18), we find that such conditions can be fulfilled only if the following condition is satisfied by the coordinate transformation (see the Appendix):
When Eq. (19) is satisfied, the condition for which the extended 2D finite embedded coordinate medium is non-reflecting can be obtained as (see the Appendix):
The condition for the second interface can be readily obtained by changing in Eq. (20) with :
Let us consider the case . Even for this case, we can still generally say that the boundaries can be made non-reflecting when the mapping function at each boundary satisfies the conditions in Eq. (20) or (23) separately and the mapping functions in between change smoothly. Under this condition, and from the duality for the TE-polarized case, we have the following form for the condition for the non-reflecting boundary:
In fact, the solution sets that satisfy Eq. (20) are not identical to those satisfying Eq. (18): in squaring Eq. (18), another set of solutions is introduced. In fact, this latter set of solutions makes the denominator of Eq. (16) or (17) zero, which physically means that these solutions are those for the excitation of the waveguide mode propagating along the y-direction with a propagation constant satisfying the squared form of Eq. (18). In practical situations, the excited waveguide modes under the condition of Eq. (21) are surface modes (formed only when the two media (I and II) have opposite signs for permittivity or permeability) which can be excited by the incidence of evanescent waves. Therefore, if we exclude this specific case of surface wave excitation that requires both opposite signs and the incidence of evanescent waves, the conditions in Eq. (24) can be regarded as necessary and sufficient conditions for a non-reflecting boundary. Otherwise, these conditions become necessary ones to make the finite-embedded CTM non-reflecting.
In Eq. (24), the ±sign means that the signs of the material parameters between the transformed and the background media can be freely selectable. However, as stated above, choosing opposite signs can result in the excitation of the surface waves at the interface, and if it is not inevitable, selecting the same sign as the surrounding media [i.e., choosing + in Eq. (24)] is recommended.
Equation (24) can be used as a guideline for the design of a planar finite-embedded CTM interface. Since the differential equation given in Eq. (24) is quite general, providing a general solution is difficult. Instead, we will explore several simple test solutions and show the utility and validity of the derived conditions in the next sections.
3. Two-dimensional examples
The simplest example satisfying Eq. (19) is the case where the transformation is totally 2D:
For this case, we can assume that the mapping functions given in Eq. (1) can be expressed in the following forms:Eqs. (26)–(28).
3.1. Beam shifter and beam expander
The beam shifter and expander based on coordinate transformation optics are shown schematically in Fig. 2 .
The mapping function of the beam shifter in its simple form can be found in Ref. , and it is given as follows:Eq. (31) being zero. Substituting these in Eq. (29) gives
For the case of , this result supports the widely-known fact that if the impedance of the medium in the virtual space for the given polarization is matched to those of the background media in the physical space, the shift transformation causes no reflection at the boundary .
Let us turn our attention to the beam expander. The simplest form of the mapping function for the beam expander can be given as Eq. (29) gives
From the definition that is a constant, the above result implies that, if the square of the magnification factor at the boundaries becomes different from , the coordinate expansion in the transversal direction (along y-direction) will cause reflection. Therefore, to make the interfaces non-reflecting, regions I and III should be filled with material that can satisfy Eq. (36).
3.2. Rotated and skewed-and-expanded transformation
Now let us consider another example in which the mapping functions in Eqs. (26)–(28) have the form of the combined characteristics of the longitudinal and transversal expansions. We will examine a simple mapping function given asEqs. (26)–(28), we have the following:Eq. (29) we have
Initially, consider the case in which and . For this case, Eq. (39) can be expressed as
Therefore, from the assumption of non-zero constants, we have . This condition can be generally satisfied by setting and . Hence the mapping in Eq. (37) can be expressed asEq. (5) because for this transformation is a unit diagonal matrix.Eq. (39).
This coordinate transformation given in Eq. (42) maps the geometrical grids of the virtual space in the Cartesian coordinates into skewed and expanded ones as presented in Fig. 3 . In Fig. 3, the dotted and circled lines correspond to lines that are parallel to the x-axis and y-axis before the transformation, respectively, and the thin grid lines denote the grid lines in the -coordinates.
In Fig. 4, we present the electric field distributions obtained from a commercial software program based on the finite element method . In this figure, the field radiated from a point (line) source propagates through a finite coordinate transformed slab which satisfies the condition of Eq. (39) by the mapping function in Eq. (42) without reflections. In this calculation, the thickness of the coordinate transformed slab is , where λ is the free-space wavelength.
3.3. Further examples
Some interesting discussions on the non-reflecting condition of the finite-embedded CTM has been reported . In that study, the following mapping function was considered:Eqs. (26)-(28) asEq. (29), we can obtain the following condition for a non-reflecting boundary:17].
Let us consider another example similar to Eq. (44) with the following form of transformation:Eqs. (26)-(28) asEq. (29), we have the condition of the non-reflecting boundary asEqs. (44) and (48), it can be found that mapping functions which enable transparency can have various other forms than that introduced in Ref. .
4. Extended example: an extended two-dimensional case
4.1. Non-reflecting beam expander
Now let us consider more complicated cases which satisfy Eq. (19). What we examine is the case in which the mapping functions are given as
This satisfies Eq. (19) for . The mapping functions in Eq. (52) correspond to the case in which the scaling of the axes is applied simultaneously in two tangential directions. For this case, the material tensor is given as
If we assume that , Eq. (22) is applicable to this case, and we arrive at the following condition for the non-reflecting boundaries:
The electric field distributions and the corresponding values of the material parameters are shown in Fig. 5.
The results presented in Eq. (54) and in Fig. 5 are contradictory to conventional knowledge on coordinate-transformation-based beam expander. Until now, it was generally believed that a non-reflecting beam expander cannot be produced except for the case where the indices of the surrounding media are selected so as to match the square value of the magnification factor at the boundaries (See the related discussions in Ref. .). However, as evidenced by Eq. (54), this is not true if we restrict the magnification to be applied in only one of the transversal directions (along the direction of y-axis in this case) and ignore the conical incidences ().
In Fig. 5, we assume that all the untransformed media are free-space () and the x-dependent magnification factor is a linear function given in Eq. (34) with the thickness of the slab d of 2λ and the magnification factor at the second interface M of 1.5. The calculated field distribution shows that the slab for the beam expander expands the beam, with almost no reflection, at the interfaces. The small observable ripples in the field distribution at the left-side of the CMT slab are due to numerical errors and the non-optimized nature of the gradient along the x-direction in the scale function . If the material parameters of the transformed region in the x-direction are carefully designed to implement adiabatic variation over a sufficiently long width, the ripples can be decreased. However, the results clearly show that the proposed method can be readily applied to the design of non-reflecting CTM.
A general method is proposed to create a non-reflecting finite-embedded coordinate transformed slab, when the mapping function is defined in extended 2D conventions under the condition of . A governing equation that can be used as the determining equation or as a guideline for the design is also proposed. To verify our approach and demonstrate the validity of our strategy, we explored several examples and discussed their topological meanings. We hope that our approach and the discussions contained in this report will be helpful in designing or understanding finite-embedded coordinate transformed media. We also hope that the approach introduced in this study can serve as a guideline or a starting point for investigations of more general cases such as conical incidences or full 3D mapping functions.
Here we show that Eq. (19) is a necessary condition for a slab of finite-embedded coordinate transformation to be non-reflecting under the constraint of the extended 2D mapping functions given in Eq. (2).
Let us start from the condition which makes the reflectance r zero in Eq. (18):
To make the interface non-reflecting, Eq. (A1) should be held for arbitrary values of . For this objective, we square both sides of Eq. (A1) and eliminate all and dependencies by using the dispersion relations for media I and II. The dispersion relation for medium I is expressed as
and that for medium II is expressed as Eq. (11), which can be re-expressed as
For any real-valued coordinate transformation cannot be zero. Therefore, we have . If we assume that the coordinate transformation can be complex-valued, and hence admit those cases of , Eq. (A3) can be expressed as
Let us examine Eq. (A4) further. When , cannot be determined. Otherwise, we have . All these cases are contrary to the claim that Eq. (A1) holds regardless of . Therefore, we discard the case where is zero. Therefore, we can express as a function of :
For the sake of simplicity, we take a simplified from of Eq. (A6) by adopting proper substitutions in the coefficients:
which can be expressed as
We square both sides of Eq. (A8) to obtain
This should be held for arbitrary values of . Therefore, we have the following set of equations that should be satisfied simultaneously:
Before solving the above coupled equations, let us first examine Eq. (A11). From this equation, we can find that either C or D should be zero. By comparing Eqs. (A6) and (A7), we can find that Eq. (A11) can be re-expressed in a more explicit form as
where the two equations correspond to the conditions that or , in the order of appearance.
Let us consider the latter case first. As we examined before, is not zero. Then if , all dependent terms in the dispersion relation of Eq. (A3) vanish, i.e., the value of is undetermined. Therefore we can also discard this case. Therefore, to satisfy Eq. (A11), we find that . With this result, we further find from Eqs. (A10) and (A12) that and .
Now, let us consider the condition of in more detail. As we examined in Eq. (A13), this corresponds to the case . This condition can be expressed using the Jacobian elements of the coordinate transformation:
With Eq. (A18), we derive Eq. (19). Equation (A18) is a necessary condition for the boundary of an extended 2D finite embedded coordinate medium to be non-reflecting. As a result, Eqs. (A17) and (A18) are necessary and sufficient conditions for an extended 2D finite embedded coordinate medium to be non-reflecting.
The authors wish to acknowledge the support of the National Research Foundation and the Ministry of Education, Science and Technology of Korea through the Creative Research Initiative Program (Active Plasmonics Application Systems).
References and links
2. A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996). [CrossRef]
3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]
5. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100(2), 024301 (2008). [CrossRef] [PubMed]
9. M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
11. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]
12. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
13. W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).
14. L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(6), 063835 (2009). [CrossRef]
15. I.-M. Lee, “Study on the transmission characteristics of the optical waves in photonic metamaterials,” PhD Dissertation (School of Electrical Engineering, Seoul National University, Seoul, Korea, 2009).
16. P. Zhang, Y. Jin, and S. He, “Inverse transformation optics and reflection analysis for two-dimensional finite embedded coordinate transformation,” arXiv:0906.2038v2 (2009).
17. I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010). [CrossRef]
18. “Comsol multiphysics” (Comsol AB), <http://www.comsol.com>.